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Generalized ulamhyers stability of C*Ternary algebra nHomomorphisms for a functional equation
Journal of Inequalities and Applications volume 2011, Article number: 30 (2011)
Abstract
In this article, we investigate the UlamHyers stability of C*ternary algebra nhomomorphisms for the functional equation:
in C*ternary algebras.
2000 Mathematics Subject Classification: Primary 39B82; 46B03; 47Jxx.
1. Introduction and preliminaries
Ternary algebraic operations were considered in the nineteenth century by several mathematicians, such as Cayley [1] who introduced the notion of cubic matrix, which, in turn, was generalized by Kapranov et al. [2]. The simplest example of such nontrivial ternary operation is given by the following composition rule:
Ternary structures and their generalization, the socalled narray structures, raise certain hopes in view of their applications in physics. Some significant physical applications are as follows (see [3]):

(1)
The algebra of nonions generated by two matrices
was introduced by Sylvester as a ternary analog of Hamilton's quaternions [4].

(2)
The quark model inspired a particular brand of ternary algebraic systems. The socalled Nambu mechnics is based on such structures [5].
There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and YangBaxter equation [4, 6].
A C*ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) α [x, y, z] of A^{3} into A, which is ℂlinear in the outer variables, conjugate ℂlinear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies [x, y, z] ≤ x · y · z and [x, x, x] = x^{3} (see [7, 8]). Every left Hilbert C*module is a C*ternary algebra via the ternary product [x, y, z] := 〈x, y〉 z.
If a C*ternary algebra (A,[·, ·, ·]) has an identity, i.e., an element e ∈ A such that x = [x, e, e] = [e, e, x] for all x ∈ A, then it is customary to verify that A, endowed with x ∘ y := [x, e, y] and x* := [e, x, e], is a unital C*algebra. Conversely, if (A, ∘) is a unital C*algebra, then [x, y, z] := x ∘ y* ∘ z makes A into a C*ternary algebra.
Let A and B be C*ternary algebras. A ℂlinear mapping H : A → B is called a C*ternary algebra homomorphism if
for all x, y, z ∈ A.
Definition. Let A and B be C*ternary algebras. A multilinear mapping H : A^{n} → B over ℂ is called a C*ternary algebra nhomomorphism if it satisfies
for all x_{1}, y_{1}, z_{1}, · · ·, x_{ n } , y_{ n } , z_{ n } ∈ A.
In 1940, Ulam [9] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms:
We are given a group G and a metric group G' with metric ρ(·, ·). Given ε > 0, does there exist a δ > 0 such that if f : G → G' satisffies ρ(f(xy), f(x) f(y)) < δ for all x, y ∈ G, then a homomorphism h : G → G' exists with ρ(f(x), h(x)) < ε for all x ∈ G ?
In 1941, Hyers [10] gave the first partial solution to Ulam's question for the case of approximate additive mappings under the assumption that G_{1} and G_{2} are Banach spaces. Then, Aoki [11] and Bourgin [12] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [13] generalized the theorem of Hyers [10] by considering the stability problem with unbounded Cauchy differences. In 1991, Gajda [14], following the same approach as that by Rassias [13], gave an affirmative solution to this question for p > 1. It was shown by Gajda [14] as well as by Rassias and Šemrl [15], that one cannot prove a Rassiastype theorem when p = 1. Găvruta [16] obtained the generalized result of Rassias's theorem which allows the Cauchy difference to be controlled by a general unbounded function. During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the generalized HyersUlam stability to a number of functional equations and mappings, for example, CauchyJensen mappings, kadditive mappings, invariant means, multiplicative mappings, bounded n th differences, convex functions, generalized orthogonality mappings, EulerLagrange functional equations, differential equations, and NavierStokes equations. The instability of characteristic flows of solutions of partial differential equations is related to the Ulam's stability of functional equations [17][24]. On the other hand, the authors [25], Park [20] and Kim [26] have contributed studies in respect of the stability problem of ternary homomorphisms and ternary derivations.
2. Solution and stability
Let X and Y be real or complex vector spaces and n ≥ 2 an integer. For a mapping f : X^{n} → Y, consider the functional equation:
The above functional equation is rewritten as
where
We solve the general problem in vector spaces for the nadditive mappings satisfying (2.1).
Theorem 2.1. A mapping f : X^{n} → Y satisffies the equation (2.1) if and only if the mapping f is nadditive.
Proof. Assume that f satisfies (2.1). Letting x_{1} = y_{1} = · · · = x_{ n } = y_{ n } = 0 in (2.2), we get f(0, · · ·, 0) = 0. Letting y_{1} = x_{2} = y_{2} = · · · = x_{ n } = y_{ n } = 0 in (2.2), we have
for all x_{1} ∈ X. Similarly, we get
for all x_{2}, · · ·, x_{ n } ∈ X. Setting y_{1} = y_{2} = 0 and x_{3} = y_{3} = · · · = x_{ n } = y_{ n } = 0 in (2.2), we have
for all x_{1}, x_{2} ∈ X. Similarly, we get f(x_{1}, 0, x_{3}, 0, · · ·, 0) = · · · = f(0, · · ·, 0, x_{n1}, x_{ n } ) = 0 for all x_{1}, · · ·, x_{ n } ∈ X.
Continuing this process, we obtain that f(x_{1}, · · ·, x_{ n } ) = 0 for all x_{1}, · · ·, x_{ n } ∈ X with x_{ i } = 0 for some i = 1, · · ·, n. Letting y_{2} = · · · = y_{ n } = 0 in (2.2), we get the additivity in the first variable. Similarly, the additivities in the remaining variables hold.
The converse is obvious. □
We investigate the generalized Ulam's stability in C*ternary algebras for the nadditive mappings satisfying (2.1).
Lemma 2.2. Let X and Y be complex vector spaces and let f : X^{n} → Y be a nadditive mapping such that
for alland all x_{1}, · · ·, x_{ n } ∈ X, then f is nlinear over ℂ.
Proof. Since f is nadditive, we get for all x_{1}, · · ·, x_{ n } ∈ X. Now let and M be an integer greater than 2(σ_{1} + · · · + σ_{ n } ). Since , there is such that . Now
for some . Thus, we have
for all x_{1}, · · ·, x_{ n } ∈ X. Hence, the mapping f : X^{n} → Y is nlinear over ℂ. □
Using the above lemma, one can obtain the following result.
Theorem 2.3. Let X and Y be complex vector spaces and let f : X^{n} → Y be a mapping such that
for alland all x_{1,1}, x_{2,1}, · · ·, x_{1,n}, x_{2,n}∈ X. Then, f is nlinear over ℂ.
Proof. Letting λ_{1} = · · · = λ_{ n } = 1, by Theorem 1.1, f is nadditive. Letting x_{2,1} = · · · = x_{2,n}= 0 in (2.3), we get f(λ_{1}x_{1}, · · ·, λ_{ n }x_{ n } ) = λ_{1} · · · λ_{ n }f(x_{1}, · · ·, x_{ n } ) for all and all x_{1}, · · ·, x_{ n } ∈ X. Hence, by Lemma 2.2, the mapping f is nlinear over ℂ. □
From now on, assume that A is a C*ternary algebra with norm  ·  _{ A } and that B is a C*ternary algebra with norm  · _{ B }.
For a given mapping f : A^{n} → B, we define
for all and all x_{1,1}, x_{2,1}, · · ·, x_{1,n}, x_{2,n}∈ A.
We prove the generalized UlamHyers stability of homomorphisms in C*ternary algebras for the functional equation
Theorem 2.4. Let p_{1}, ···, p_{ n } ∈ (0, ∞) withand θ ∈ (0, ∞), and let f : A^{n} → B be a mapping such that
and
for alland all x_{1}, y_{1}, z_{1}, · · ·, x_{ n } , y_{ n } , z_{ n } ∈ A. Then, there exists a unique C*ternary algebra nhomomorphism H : A^{n} → B such that
for all x_{1}, · · ·, x_{ n } ∈ A.
Proof. Letting λ_{1} = · · · = λ_{ n } = 1, y_{1} = x_{1}, · · ·, y_{ n } = x_{ n } in (2.4), we gain
for all x_{1}, · · ·, x_{ n } ∈ A. Thus, we have
for all x_{1}, · · ·, x_{ n } ∈ A and all j ∈ ℕ. For given integer l, m(0 ≤ l < m), we obtain that
for all x_{1}, · · ·, x_{ n } ∈ A. Since , the sequence
is a Cauchy sequence for all x_{1}, ···, x_{ n } ∈ A. Since B is complete, the sequence converges for all x_{1}, · · ·, x_{ n } ∈ A. Define H : A^{n} → B by
for all x_{1}, · · ·, x_{ n } ∈ A. Letting l = 0 and taking m → ∞ in (2.8), one can obtain the inequality (2.6). By (2.4), we see that
for all x_{1}, y_{1}, · · ·, x_{ n } , y_{ n } ∈ A and all s. Since , letting s → ∞ in the above inequality, H satisfies (2.1). By Theorem 2.1, H is nadditive.
Letting y_{1} = x_{1}, · · ·, y_{ n } = x_{ n } in (2.4), we gain
for all and all x_{1}, · · ·, x_{ n } ∈ A. Thus we have
for all , all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ. Hence, we get
for all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ, and one can show that
for all , all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ. Hence,
for all , all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ. Since , we have
as m → ∞ for all and all x_{1}, · · ·, x_{ n } ∈ A. Hence
for all and all x_{1}, ···, x_{ n } ∈ A. From Lemma 2.2, the mapping H : A^{n} → B is nlinear over ℂ. It follows from (2.5) that
for all x_{1}, y_{1}, z_{1}, ···, x_{ n } , y_{ n } , z_{ n } ∈ A. So
for all x_{1}, y_{1}, z_{1}, ···, x_{ n } , y_{ n } , z_{ n } ∈ A.
Now, let T : A^{n} → B be another nadditive mapping satisfying (2.6). Then, we have
which tends to zero as m → ∞ for all x_{1}, ···, x_{ n } ∈ A. Hence, we can conclude that H(x_{1}, ···, x_{ n } ) = T(x_{1}, ···, x_{ n } ) for all x_{1}, ···, x_{ n } ∈ A. This proves the uniqueness of H.
Thus, the mapping H : A→B is a unique C*ternary algebra nhomomorphism satisfying (2.6). □
Letting p_{1} = ··· = p_{ n } = 0 and θ = ε in Theorem 2.4, we obtain the UlamHyers stability for the nadditive functional equation (2.1).
Corollary 2.5. Let ε ∈ (0, ∞) and let f : A^{n} → B be a mapping satisfying
and
for alland all x_{1}, y_{1}, z_{1} ···, x_{ n } , y_{ n } , z_{ n } ∈ A. Then, there exists a unique C*ternary algebra nhomomorphism H : A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Example 2.6. We present the following counterexample modiffied by the wellknown counterexample of Z. Gajda[14]for the functional equation (2.1). Set θ > 0 and let.
Deffine a function f :ℝ ^{n} → ℝ by
for all x_{1}, ···, x_{ n } ∈ ℝ, where ϕ_{ μ } : ℝ ^{n} → ℝ is the function given by
for all x_{1}, ···, x_{ n } ∈ ℝ. Deffine another function g : ℝ → ℝ by
for all x ∈ ℝ.
For all m ∈ ℕ and all, we assert that
It was proved in[14]that
for all x, y ∈ ℝ, that is, (2.9) holds for m = 1. For a ffixed k ∈ ℕ, assume that (2.9) holds for m = k. Then, we have
for all, that is, (2.9) holds for m = k + 1. Hence, (2.9) holds for all m ∈ ℝ.
Note that
for all x_{1}, ···, x_{ n }∈ ℝ. By the inequality (2.9) and the above equality, we see that
for all x_{1}, y_{1}, ···, x_{ n } , y_{ n } ∈ ℝ. However, we observe from[14]that
and so
Thus,
where h: ℝ^{n} → ℝ is the function given by
for all x_{1}, ···, x_{ n }∈ ℝ. Hence, the function f is a counterexample for the singular caseof Theorem 2.4.
Theorem 2.7. Let p ∈ (0,n) and θ ∈ (0, ∞), and let f : A^{n} → B be a mapping such that
and
for alland all x_{1}, y_{1}, z_{1} ···, x_{ n }, y_{ n }, z_{ n }∈ A. Then, there exists a unique C*ternary algebra nhomomorphism H : A^{n}→B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to the proof of Theorem 2.4. □
Example 2.8. We present the following counterexample modiffied by the wellknown counterexample of Z. Gajda[14]for the functional equation (2.1). Set θ > 0 and let
Let f : ℝ^{n} → ℝ and g : ℝ → ℝ be the same as in Example 2.6. By the same argument as in Example 2.6, for all m ∈ ℕ and all, one can obtain that g satisffies the inequality:
By the equality (2.10) and the above inequality, we see that
for all x_{1}, y_{1}, ···, x_{ n } , y_{ n }∈ ℝ. For each x_{1}, y_{1}, ···, x_{ n } , y_{ n }∈ ℝ, let M(x_{1}, y_{1}, ···, x_{ n } , y_{ n }) := max{x_{1}, y_{1}, ···, x_{ n } , y_{ n } }. We have
for all x_{1}, y_{1}, ···, x_{ n } , y_{ n }∈ ℝ. Thus we have
for all x_{1}, y_{1}, ···, x_{ n }, y_{ n }∈ ℝ. By the same reason as for Example 2.6, the function f is a counterexample for the singular case p = n of Theorem 2.7.
Theorem 2.9. Let p_{1}, ···, p_{ n } ∈ (0, ∞) with, s ∈ (0, n) and θ, η ∈ (0, ∞), and let f: A^{n} → B be a mapping such that
and
for alland all x_{1,1}, x_{2,1}, x_{3,1}, ···, x_{1, n}, x_{2, n}, x_{3, n}∈ A. Then, there exists a unique C*ternary algebra nhomomorphism H: A^{n}→ B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to the proof of Theorem 2.4. □
Theorem 2.10. Let p_{1}, ···, p_{ n } ∈ (0, ∞) withand θ ∈ (0, ∞), and let f : A^{n} → B be a mapping satisfying (2.4) (2.5). Then, there exists a unique C*ternary algebra nhomomorphism H : A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. It follows from (2.7) that
for all x_{1}, ···, x_{ n } ∈ A. Hence,
for all nonnegative integers m and l with m > l and all x_{1}, ···, x_{ n } ∈ A. It follows from (2.16) that the sequence is a Cauchy sequence for all x_{1}, ···, x_{ n } ∈ A. Since B is complete, the sequence converges. Hence, one can define the mapping H : A^{n} → B by for all x_{1}, ···, x_{ n } ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (2.16), we get (2.15).
The remainder of the proof is similar to the proof of Theorem 2.4. □
Theorem 2.11. Let p ∈ (3n, ∞) and θ ∈ (0, ∞), and let f: A^{n} → B be a mapping satisfying (2.11) (2.7), and f(0, ···, 0) = 0. Then, there exists a unique C*ternary algebra nhomomorphism H : A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to that of Theorem 2.10. □
Theorem 2.12. Let p_{1}, ···, p_{ n } ∈ (0, ∞) with, s ∈ (n, ∞) and θ, η ∈ (0, ∞), and let f: A^{n} → B be a mapping such that (2.13), (2.14), and f(0, ···, 0) = 0. Then, there exists a unique C*ternary algebra nhomomorphism H: A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to that of Theorem 2.10.
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The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.
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Park, W., Bae, J. Generalized ulamhyers stability of C*Ternary algebra nHomomorphisms for a functional equation. J Inequal Appl 2011, 30 (2011). https://doi.org/10.1186/1029242X201130
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Keywords
 nhomomorphisms
 C*ternary algebra